# Verification checks (P4)

1. Reproducibility: deleted out/results.json, re-ran analysis.py, cmp -> byte-identical. PASS
2. Closed form satisfies Bellman: brute-force maximisation of ln c + beta*(A + Balphaln(1-c)) at y=1 over 2,000,001 points gives max -19.5244122217225 vs V(1) = A = -19.5244122217224, argmax c = 0.658000 = (1-alpha*beta). PASS
3. max_abs_error_policy recomputed from the three c_num/c_closed pairs = 0.0063750 (matches). PASS
4. VFI converged: final_sup_change 9.997e-9 < tol 1e-8, n_iterations 361; theoretical error bound beta/(1-beta)*final change = 1.9e-7 utils. PASS
5. Policy never at grid bounds (diag_policy_hits_lower/upper_bound = 0): no truncation. PASS
6. Magnitudes vs baseline: VFI implied savings rates 1 - c_num/y = 0.3424, 0.3410, 0.3452 vs closed-form 0.342; value error at y=1 = -3.4e-4 utils, consistent with discrete-choice discretisation (VFI value <= true value is expected since the choice set is restricted; observed negative). PASS
7. Counts: n_input_rows 10 = rows in model.csv; n_analysed = grid_points_used = 400. PASS